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Theorem cantnfres 9116
 Description: The CNF function respects extensions of the domain to a larger ordinal. (Contributed by Mario Carneiro, 25-May-2015.)
Hypotheses
Ref Expression
cantnfs.s 𝑆 = dom (𝐴 CNF 𝐵)
cantnfs.a (𝜑𝐴 ∈ On)
cantnfs.b (𝜑𝐵 ∈ On)
cantnfrescl.d (𝜑𝐷 ∈ On)
cantnfrescl.b (𝜑𝐵𝐷)
cantnfrescl.x ((𝜑𝑛 ∈ (𝐷𝐵)) → 𝑋 = ∅)
cantnfrescl.a (𝜑 → ∅ ∈ 𝐴)
cantnfrescl.t 𝑇 = dom (𝐴 CNF 𝐷)
cantnfres.m (𝜑 → (𝑛𝐵𝑋) ∈ 𝑆)
Assertion
Ref Expression
cantnfres (𝜑 → ((𝐴 CNF 𝐵)‘(𝑛𝐵𝑋)) = ((𝐴 CNF 𝐷)‘(𝑛𝐷𝑋)))
Distinct variable groups:   𝐵,𝑛   𝐷,𝑛   𝐴,𝑛   𝜑,𝑛
Allowed substitution hints:   𝑆(𝑛)   𝑇(𝑛)   𝑋(𝑛)

Proof of Theorem cantnfres
Dummy variables 𝑘 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cantnfrescl.d . . . . . . . . . . . . 13 (𝜑𝐷 ∈ On)
2 cantnfrescl.b . . . . . . . . . . . . 13 (𝜑𝐵𝐷)
3 cantnfrescl.x . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (𝐷𝐵)) → 𝑋 = ∅)
41, 2, 3extmptsuppeq 7829 . . . . . . . . . . . 12 (𝜑 → ((𝑛𝐵𝑋) supp ∅) = ((𝑛𝐷𝑋) supp ∅))
5 oieq2 8953 . . . . . . . . . . . 12 (((𝑛𝐵𝑋) supp ∅) = ((𝑛𝐷𝑋) supp ∅) → OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) = OrdIso( E , ((𝑛𝐷𝑋) supp ∅)))
64, 5syl 17 . . . . . . . . . . 11 (𝜑 → OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) = OrdIso( E , ((𝑛𝐷𝑋) supp ∅)))
76fveq1d 6645 . . . . . . . . . 10 (𝜑 → (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘) = (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘))
873ad2ant1 1130 . . . . . . . . 9 ((𝜑𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) ∧ 𝑧 ∈ On) → (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘) = (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘))
98oveq2d 7146 . . . . . . . 8 ((𝜑𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) ∧ 𝑧 ∈ On) → (𝐴o (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)) = (𝐴o (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)))
10 suppssdm 7818 . . . . . . . . . . . . 13 ((𝑛𝐵𝑋) supp ∅) ⊆ dom (𝑛𝐵𝑋)
11 eqid 2821 . . . . . . . . . . . . . . 15 (𝑛𝐵𝑋) = (𝑛𝐵𝑋)
1211dmmptss 6068 . . . . . . . . . . . . . 14 dom (𝑛𝐵𝑋) ⊆ 𝐵
1312a1i 11 . . . . . . . . . . . . 13 (𝜑 → dom (𝑛𝐵𝑋) ⊆ 𝐵)
1410, 13sstrid 3954 . . . . . . . . . . . 12 (𝜑 → ((𝑛𝐵𝑋) supp ∅) ⊆ 𝐵)
15143ad2ant1 1130 . . . . . . . . . . 11 ((𝜑𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) ∧ 𝑧 ∈ On) → ((𝑛𝐵𝑋) supp ∅) ⊆ 𝐵)
16 eqid 2821 . . . . . . . . . . . . . 14 OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) = OrdIso( E , ((𝑛𝐵𝑋) supp ∅))
1716oif 8970 . . . . . . . . . . . . 13 OrdIso( E , ((𝑛𝐵𝑋) supp ∅)):dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅))⟶((𝑛𝐵𝑋) supp ∅)
1817ffvelrni 6823 . . . . . . . . . . . 12 (𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) → (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘) ∈ ((𝑛𝐵𝑋) supp ∅))
19183ad2ant2 1131 . . . . . . . . . . 11 ((𝜑𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) ∧ 𝑧 ∈ On) → (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘) ∈ ((𝑛𝐵𝑋) supp ∅))
2015, 19sseldd 3944 . . . . . . . . . 10 ((𝜑𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) ∧ 𝑧 ∈ On) → (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘) ∈ 𝐵)
2120fvresd 6663 . . . . . . . . 9 ((𝜑𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) ∧ 𝑧 ∈ On) → (((𝑛𝐷𝑋) ↾ 𝐵)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)) = ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)))
2223ad2ant1 1130 . . . . . . . . . . 11 ((𝜑𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) ∧ 𝑧 ∈ On) → 𝐵𝐷)
2322resmptd 5881 . . . . . . . . . 10 ((𝜑𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) ∧ 𝑧 ∈ On) → ((𝑛𝐷𝑋) ↾ 𝐵) = (𝑛𝐵𝑋))
2423fveq1d 6645 . . . . . . . . 9 ((𝜑𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) ∧ 𝑧 ∈ On) → (((𝑛𝐷𝑋) ↾ 𝐵)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)) = ((𝑛𝐵𝑋)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)))
258fveq2d 6647 . . . . . . . . 9 ((𝜑𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) ∧ 𝑧 ∈ On) → ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)) = ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)))
2621, 24, 253eqtr3d 2864 . . . . . . . 8 ((𝜑𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) ∧ 𝑧 ∈ On) → ((𝑛𝐵𝑋)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)) = ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)))
279, 26oveq12d 7148 . . . . . . 7 ((𝜑𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) ∧ 𝑧 ∈ On) → ((𝐴o (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐵𝑋)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘))) = ((𝐴o (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘))))
2827oveq1d 7145 . . . . . 6 ((𝜑𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) ∧ 𝑧 ∈ On) → (((𝐴o (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐵𝑋)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘))) +o 𝑧) = (((𝐴o (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘))) +o 𝑧))
2928mpoeq3dva 7205 . . . . 5 (𝜑 → (𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴o (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐵𝑋)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘))) +o 𝑧)) = (𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴o (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘))) +o 𝑧)))
306dmeqd 5747 . . . . . 6 (𝜑 → dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) = dom OrdIso( E , ((𝑛𝐷𝑋) supp ∅)))
31 eqid 2821 . . . . . 6 On = On
32 mpoeq12 7201 . . . . . 6 ((dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) = dom OrdIso( E , ((𝑛𝐷𝑋) supp ∅)) ∧ On = On) → (𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴o (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘))) +o 𝑧)) = (𝑘 ∈ dom OrdIso( E , ((𝑛𝐷𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴o (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘))) +o 𝑧)))
3330, 31, 32sylancl 589 . . . . 5 (𝜑 → (𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴o (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘))) +o 𝑧)) = (𝑘 ∈ dom OrdIso( E , ((𝑛𝐷𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴o (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘))) +o 𝑧)))
3429, 33eqtrd 2856 . . . 4 (𝜑 → (𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴o (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐵𝑋)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘))) +o 𝑧)) = (𝑘 ∈ dom OrdIso( E , ((𝑛𝐷𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴o (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘))) +o 𝑧)))
35 eqid 2821 . . . 4 ∅ = ∅
36 seqomeq12 8065 . . . 4 (((𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴o (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐵𝑋)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘))) +o 𝑧)) = (𝑘 ∈ dom OrdIso( E , ((𝑛𝐷𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴o (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘))) +o 𝑧)) ∧ ∅ = ∅) → seqω((𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴o (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐵𝑋)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘))) +o 𝑧)), ∅) = seqω((𝑘 ∈ dom OrdIso( E , ((𝑛𝐷𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴o (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘))) +o 𝑧)), ∅))
3734, 35, 36sylancl 589 . . 3 (𝜑 → seqω((𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴o (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐵𝑋)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘))) +o 𝑧)), ∅) = seqω((𝑘 ∈ dom OrdIso( E , ((𝑛𝐷𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴o (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘))) +o 𝑧)), ∅))
3837, 30fveq12d 6650 . 2 (𝜑 → (seqω((𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴o (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐵𝑋)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘))) +o 𝑧)), ∅)‘dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅))) = (seqω((𝑘 ∈ dom OrdIso( E , ((𝑛𝐷𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴o (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘))) +o 𝑧)), ∅)‘dom OrdIso( E , ((𝑛𝐷𝑋) supp ∅))))
39 cantnfs.s . . 3 𝑆 = dom (𝐴 CNF 𝐵)
40 cantnfs.a . . 3 (𝜑𝐴 ∈ On)
41 cantnfs.b . . 3 (𝜑𝐵 ∈ On)
42 cantnfres.m . . 3 (𝜑 → (𝑛𝐵𝑋) ∈ 𝑆)
43 eqid 2821 . . 3 seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐵𝑋)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘))) +o 𝑧)), ∅) = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐵𝑋)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘))) +o 𝑧)), ∅)
4439, 40, 41, 16, 42, 43cantnfval2 9108 . 2 (𝜑 → ((𝐴 CNF 𝐵)‘(𝑛𝐵𝑋)) = (seqω((𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴o (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐵𝑋)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘))) +o 𝑧)), ∅)‘dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅))))
45 cantnfrescl.t . . 3 𝑇 = dom (𝐴 CNF 𝐷)
46 eqid 2821 . . 3 OrdIso( E , ((𝑛𝐷𝑋) supp ∅)) = OrdIso( E , ((𝑛𝐷𝑋) supp ∅))
47 cantnfrescl.a . . . . 5 (𝜑 → ∅ ∈ 𝐴)
4839, 40, 41, 1, 2, 3, 47, 45cantnfrescl 9115 . . . 4 (𝜑 → ((𝑛𝐵𝑋) ∈ 𝑆 ↔ (𝑛𝐷𝑋) ∈ 𝑇))
4942, 48mpbid 235 . . 3 (𝜑 → (𝑛𝐷𝑋) ∈ 𝑇)
50 eqid 2821 . . 3 seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘))) +o 𝑧)), ∅) = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘))) +o 𝑧)), ∅)
5145, 40, 1, 46, 49, 50cantnfval2 9108 . 2 (𝜑 → ((𝐴 CNF 𝐷)‘(𝑛𝐷𝑋)) = (seqω((𝑘 ∈ dom OrdIso( E , ((𝑛𝐷𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴o (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘))) +o 𝑧)), ∅)‘dom OrdIso( E , ((𝑛𝐷𝑋) supp ∅))))
5238, 44, 513eqtr4d 2866 1 (𝜑 → ((𝐴 CNF 𝐵)‘(𝑛𝐵𝑋)) = ((𝐴 CNF 𝐷)‘(𝑛𝐷𝑋)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2115  Vcvv 3471   ∖ cdif 3907   ⊆ wss 3910  ∅c0 4266   ↦ cmpt 5119   E cep 5437  dom cdm 5528   ↾ cres 5530  Oncon0 6164  ‘cfv 6328  (class class class)co 7130   ∈ cmpo 7132   supp csupp 7805  seqωcseqom 8058   +o coa 8074   ·o comu 8075   ↑o coe 8076  OrdIsocoi 8949   CNF ccnf 9100 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rmo 3134  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-se 5488  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6121  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7088  df-ov 7133  df-oprab 7134  df-mpo 7135  df-om 7556  df-1st 7664  df-2nd 7665  df-supp 7806  df-wrecs 7922  df-recs 7983  df-rdg 8021  df-seqom 8059  df-oadd 8081  df-er 8264  df-map 8383  df-en 8485  df-dom 8486  df-sdom 8487  df-fin 8488  df-fsupp 8810  df-oi 8950  df-cnf 9101 This theorem is referenced by: (None)
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